Engaging students: Mathematical induction

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Emily Bruce. Her topic, from Precalculus: mathematical induction.

How could you as a teacher create an activity or project that involves your topic?

In order to help students understand how induction works, I would either use the domino example or the idea behind an assembly line. Using dominos, students would make a domino train by standing them on their ends close to each other. Students should be able to see that we only have to knock down the first one in order to guarantee that all of them fall over because we know that any one tile falling over will knock over the next one. The assembly line analogy uses the idea that as long as an object begins down the assembly line and each person does their job and passes it on, the object will be made correctly. Like induction, these examples only require us to have the first step succeed and guarantee that it passes from one step to the next, in order to guarantee that it will work for every step.

How can this topic be used in your students’ future courses in mathematics or science?

Induction is a basic proof method that is very useful when proving statements that involve all natural numbers. It is used in pre-calculus, as well as more advanced calculus courses and other upper level college courses. It is an extremely helpful tool when dealing with the natural numbers. Using induction, we can prove conjectures about different series or summations. Knowing and understanding different patterns of with the natural numbers is particularly important in later calculus classes when they focus on the possible convergence of different series and summations.

How has this topic appeared in pop culture?

The movie Titanic is a classic movie about a sinking cruise ship. The question to be posed is “How did they know the whole ship would sink from one hole?” The answer involves induction. The captain would have known that the bulkhead that had a hole would flood completely from the hole. He also would have known that as soon as any one bulkhead was full, another adjacent bulkhead would begin filling up. This is the concept of induction. Using just those two pieces of information, the captain was able to induce that the boat would continue to fill with water until it sank. This is why the captain immediately began evacuating the boat. It was only a matter of time before the ship went down, with everything in it. He knew all of this just form knowing that the first bulkhead would fill and once any one was full, the next would begin to fill as well. The knowledge and quick thinking of the captain saved many lives from the Titanic.